Lagmental Vicfred

Divisor-Count and Divisor-Sum Functions Factor Prime by Prime by Vicfred

Last updated: Mon 03 January 2022

tags: mathematics, arithmetic functions, Dirichlet convolution, Mobius inversion

The functions tau and sigma are multiplicative because divisors of coprime products split uniquely. I will separate the object being defined from the consequence being claimed.

Definitions first

Arithmetic functions \(f:\mathbf N\to\mathbf C\) form a commutative ring under Dirichlet convolution. Multiplicative functions are determined by their values on prime powers \(p^k\).

$$ n=\prod_pp^{a_p} $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ \tau(n)=\prod_p(a_p+1),\qquad\sigma(n)=\prod_p\frac{p^{a_p+1}-1}{p-1} $$

A small case in full

The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.

$$ 360=2^3\,3^2\,5\quad\Longrightarrow\quad\begin{cases}\tau(360)=4\cdot3\cdot2=24,\\\sigma(360)=15\cdot13\cdot6=1170.\end{cases} $$

The reusable statement

The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.

$$ \begin{aligned} \mathsf{D}\;&:\quad n=\prod_pp^{a_p},\\[5pt] \mathsf{C}\;&:\quad \tau(n)=\prod_p(a_p+1),\qquad\sigma(n)=\prod_p\frac{p^{a_p+1}-1}{p-1}. \end{aligned} $$

A nearby false statement

Pointwise multiplication and Dirichlet convolution are different operations. Möbius inversion reverses convolution with the constant-one function, not ordinary multiplication.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \tau(n)=\prod_p(a_p+1),\qquad\sigma(n)=\prod_p\frac{p^{a_p+1}-1}{p-1} \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Tue 13 September 2016. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.